# Famous Figures and Diagrams in Economics

## Edited by Mark Blaug and Peter Lloyd

# Chapter 27: The Logistic Growth Curve

## J.S. Cramer

## Extract

J. S. Cramer The logistic curve was devised in the 19th century to describe the time path of a growing human population. It was first proposed (and named) by Verhulst, a pupil of the Belgian statistician and demographer Quetelet, forgotten until the 1920s, and is now mainly used as an instrument of market research. In order to describe the course over time of a quantity W(t) consider # its rate of change W = dW(t)/dt. The simplest assumption is that this is # proportional to the size already attained, or W(t) = b W(t), which leads of course to exponential growth, as in Malthus’s famous dictum that population, if unchecked, will increase at a geometric rate. For the greater part of the nineteenth century, this was not a bad approximation for the immigrant population of continents like Northern America and Australia, yet it was clear that the process could not continue indefinitely. Verhulst (1838, 1845) therefore modified the differential equation by introducing an upper limit or saturation level Ω, writing # W(t) = b W(t) (Ω 2 W(t)) (27.1) so that the rate of growth is proportional to both the level attained and the remaining room for further expansion. If we consider the proportion # Y(t) = W(t) /Ω we have Y(t) = b Y(t) (1 2 Y(t)) and the solution of this differential equation is Y(t)= exp(a + bt)/{1 + exp(a + bt)} or Y(z) = exp z/...

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